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GEOMETRIC APPROACH TO THE HAMILTON–JACOBI EQUATION AND GLOBAL PARAMETRICES FOR THE SCHRÖDINGER PROPAGATOR

SANDRO GRAFFI

Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy

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and

LORENZO ZANELLI

Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy

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https://doi.org/10.1142/S0129055X11004497Cited by:6 (Source: Crossref)

Abstract

We construct a family of global Fourier Integral Operators, defined for arbitrary large times, representing a global parametrix for the Schrödinger propagator when the potential is quadratic at infinity. This construction is based on the geometric approach to the corresponding Hamilton–Jacobi equation and thus sidesteps the problem of the caustics generated by the classical flow. Moreover, a detailed study of the real phase function allows us to recover a WKB semiclassical approximation which necessarily involves the multivaluedness of the graph of the Hamiltonian flow past the caustics.

Keywords:

AMSC: 47D08, 35S30, 53D12

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